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Theorem fvmptdv2 5292
Description: Alternate deduction version of fvmpt 5281, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptdv2.1  |-  ( ph  ->  A  e.  D )
fvmptdv2.2  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
fvmptdv2.3  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
fvmptdv2  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
Distinct variable groups:    x, A    x, C    x, D    ph, x
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem fvmptdv2
StepHypRef Expression
1 eqidd 2083 . . 3  |-  ( ph  ->  ( x  e.  D  |->  B )  =  ( x  e.  D  |->  B ) )
2 fvmptdv2.3 . . 3  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
3 fvmptdv2.1 . . 3  |-  ( ph  ->  A  e.  D )
4 elex 2611 . . . . . 6  |-  ( A  e.  D  ->  A  e.  _V )
53, 4syl 14 . . . . 5  |-  ( ph  ->  A  e.  _V )
6 isset 2606 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
75, 6sylib 120 . . . 4  |-  ( ph  ->  E. x  x  =  A )
8 fvmptdv2.2 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  B  e.  V )
9 elex 2611 . . . . . 6  |-  ( B  e.  V  ->  B  e.  _V )
108, 9syl 14 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  B  e.  _V )
112, 10eqeltrrd 2157 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  C  e.  _V )
127, 11exlimddv 1820 . . 3  |-  ( ph  ->  C  e.  _V )
131, 2, 3, 12fvmptd 5285 . 2  |-  ( ph  ->  ( ( x  e.  D  |->  B ) `  A )  =  C )
14 fveq1 5208 . . 3  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( F `  A
)  =  ( ( x  e.  D  |->  B ) `  A ) )
1514eqeq1d 2090 . 2  |-  ( F  =  ( x  e.  D  |->  B )  -> 
( ( F `  A )  =  C  <-> 
( ( x  e.  D  |->  B ) `  A )  =  C ) )
1613, 15syl5ibrcom 155 1  |-  ( ph  ->  ( F  =  ( x  e.  D  |->  B )  ->  ( F `  A )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602    |-> cmpt 3847   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by: (None)
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